where \(h_{pq}\)  and \(h_{pqrs}\)  refer to one- and two- electron integrals, respectively.
The integrals are supplied as inputs to the VQE algorithm, using a program run on a
classical computer. In Eq. (REF ), the Hamiltonian is compactly described using a single index, \(\alpha \) , and the \(P_{\alpha }\) s no longer refer to the second
quantized creation and annihilation operators, but their mapped version into their qubit
operator form, that is, a string of tensor product of Pauli operators. This can be done by, for example, the Jordan-Wigner transformation [1]}. \(T_2(\theta )\)  can also be appropriately mapped into its qubit operator form, and this allows \(|\Psi (\theta ) \rangle \)  to be recast in its circuit form [2]}, with appropriately chosen initial guess parameters, \(\theta _{init}\) . The expectation value is calculated by either using matrix
operations (statevector (SV) backend) or the circuit is evaluated by measuring \(H\)  on
\(|\Psi (\theta _{init}) \rangle \)  (QASM backend) and the corresponding energy extracted. This sub-part constitutes the quantum module of the classical-quantum VQE approach. The energy is then minimized with respect to the parameters by using a suitable optimization algorithm on a classical computer, over a series of iterations, where in each VQE iteration, the parameter \(\theta \)  incrementally changes, and is fed back into the circuit to obtain an updated value of energy for that iteration, and the process is repeated until a minimum is found.
